On perturbation procedure for limit cycle analysis

Existence and analytical approximations of limit cycles in a three-dimensional nonlinear autonomous feedback control system

“…Concerning the higher-order results, to get better accurate approximation, in this paper, we adopt the idea of a perturbation procedure of Chen, Cheung, and Lau [8] proposed in van der Pol equation. Chen, Cheung, and Lau [8] pointed out that, in approximating limit cycles (not periodic solutions in the conservative problems), the general solutions of the homogeneous parts in the perturbation equations of each order should have the same phase. That is to say, in this paper, the general solutions of the homogeneous parts in the perturbation equations of each order should be assumed in the following form (note: taking the form of the zeroth-order approximation (10) into account):…”

A new perturbation procedure for limit cycle analysis in three-dimensional nonlinear autonomous dynamical systems

“…Following the procedure employed in the above section, a new parametric transformation α defined in (7) is introduced and the approximate formulations for the limit cycle of system (21) are assumed as (10).…”

“…Until now, various improved and novel perturbation techniques have been developed such as the modified L-P method [3], the elliptic L-P method [4], the elliptic perturbation method [5], the nonlinear time transformation method [6], the generalized averaging method [7], the nonlinear scales method [8], the modified KBM method [9], etc. In particular, Chen et al [10] pointed out that the use of conditions of constant phase angles in the perturbation procedures could provide more accurate results for limit cycle analysis, especially for the strongly nonlinear cases. Starting from different viewpoints, He [11,12] also extended the classical L-P method and developed two modified versions of the L-P methods.…”

Limit cycle analysis of a class of strongly nonlinear oscillation equations

“…Multiplying equation (9) respectively by V (1) and V(2) and integrating from 0 to 27r with respect to ¢, we obtain the solvability conditions for equation (9) as follows:…”

“…Margallo et al [6][7][8] discussed the limit cycles of the generalized van der Pol equation of the form 4i + Au -2Bu 3 + e(z3 + z2u 2 + ztu4)i* = 0 by using the Jacobian elliptic functions and the method of harmonic balance. Chen et al [9] obtained the limit cycles of the van der Pol equation by applying the two-variable expansion procedure with nonlinear scales in the method of multiple scales. Moremedi et al [10] used the derivative expansion version of the method of multiple scales to discuss the limit cycle of a generalized van der Pol equation of the form/i + u = e(1 -u2n)i*, in which n is any positive integer.…”